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52 Chapter 2 DISCRETE-TIME SIGNALS AND SYSTEMS
subject to the initial conditions:
{y(n); −N ≤ n ≤−1} and {x(n); −M ≤ n ≤−1}
A solution to (2.24) can be obtained in the form
y(n) =y ZI (n)+y ZS (n)
where y ZI (n) iscalled the zero-input solution, which is a solution due
to the initial conditions alone (assuming they exist), while the zero-state
solution, y ZS (n), is a solution due to input x(n) alone (or assuming that
the initial conditions are zero). In MATLAB another form of the function
filter can be used to solve for the difference equation, given its initial
conditions. We will illustrate the use of this form in Chapter 4.
2.4.3 DIGITAL FILTERS
Filter is a generic name that means a linear time-invariant system designed
for a specific job of frequency selection or frequency discrimination. Hence
discrete-time LTI systems are also called digital filters. There are two
types ofdigital filters.
FIR filter If the unit impulse response of an LTI system is of finite
duration, then the system is called a finite-duration impulse response (or
FIR) filter. Hence for an FIR filter h(n) =0fornn 2 .
The following part of the difference equation (2.21) describes a causal FIR
filter:
M∑
y(n) = b m x(n − m) (2.25)
m=0
Furthermore, h(0) = b 0 , h(1) = b 1 , ...,h(M) =b M , while all other h(n)’s
are 0. FIR filters are also called nonrecursive or moving average (MA)
filters. In MATLAB FIR filters are represented either as impulse response
values {h(n)} or as difference equation coefficients {b m } and {a 0 =1}.
Therefore to implement FIR filters, we can use either the conv(x,h)
function (and its modification that we discussed) or the filter(b,1,x)
function. There is a difference in the outputs of these two implementations
that should be noted. The output sequence from the conv(x,h) function
has a longer length than both the x(n) and h(n) sequences. On the other
hand, the output sequence from the filter(b,1,x) function has exactly
the same length as the input x(n) sequence. In practice (and especially
for processing signals) the use of the filter function is encouraged.
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